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Theorem ofrn2 29281
Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
ofrn.1 (𝜑𝐹:𝐴𝐵)
ofrn.2 (𝜑𝐺:𝐴𝐵)
ofrn.3 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
ofrn.4 (𝜑𝐴𝑉)
Assertion
Ref Expression
ofrn2 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Proof of Theorem ofrn2
Dummy variables 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
2 ffn 6002 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
43adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐹 Fn 𝐴)
5 simprl 793 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑎𝐴)
6 fnfvelrn 6312 . . . . . 6 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
74, 5, 6syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐹𝑎) ∈ ran 𝐹)
8 ofrn.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
9 ffn 6002 . . . . . . . 8 (𝐺:𝐴𝐵𝐺 Fn 𝐴)
108, 9syl 17 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
1110adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐺 Fn 𝐴)
12 fnfvelrn 6312 . . . . . 6 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1311, 5, 12syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐺𝑎) ∈ ran 𝐺)
14 simprr 795 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑧 = ((𝐹𝑎) + (𝐺𝑎)))
15 rspceov 6645 . . . . 5 (((𝐹𝑎) ∈ ran 𝐹 ∧ (𝐺𝑎) ∈ ran 𝐺𝑧 = ((𝐹𝑎) + (𝐺𝑎))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
167, 13, 14, 15syl3anc 1323 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
1716rexlimdvaa 3025 . . 3 (𝜑 → (∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎)) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
1817ss2abdv 3654 . 2 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
19 ofrn.4 . . . . 5 (𝜑𝐴𝑉)
20 inidm 3800 . . . . 5 (𝐴𝐴) = 𝐴
21 eqidd 2622 . . . . 5 ((𝜑𝑎𝐴) → (𝐹𝑎) = (𝐹𝑎))
22 eqidd 2622 . . . . 5 ((𝜑𝑎𝐴) → (𝐺𝑎) = (𝐺𝑎))
233, 10, 19, 19, 20, 21, 22offval 6857 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
2423rneqd 5313 . . 3 (𝜑 → ran (𝐹𝑓 + 𝐺) = ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
25 eqid 2621 . . . 4 (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2625rnmpt 5331 . . 3 ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))}
2724, 26syl6eq 2671 . 2 (𝜑 → ran (𝐹𝑓 + 𝐺) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))})
28 ofrn.3 . . . . 5 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
29 ffn 6002 . . . . 5 ( + :(𝐵 × 𝐵)⟶𝐶+ Fn (𝐵 × 𝐵))
3028, 29syl 17 . . . 4 (𝜑+ Fn (𝐵 × 𝐵))
31 frn 6010 . . . . . 6 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
321, 31syl 17 . . . . 5 (𝜑 → ran 𝐹𝐵)
33 frn 6010 . . . . . 6 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
348, 33syl 17 . . . . 5 (𝜑 → ran 𝐺𝐵)
35 xpss12 5186 . . . . 5 ((ran 𝐹𝐵 ∧ ran 𝐺𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
3632, 34, 35syl2anc 692 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
37 ovelimab 6765 . . . 4 (( + Fn (𝐵 × 𝐵) ∧ (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵)) → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3830, 36, 37syl2anc 692 . . 3 (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3938abbi2dv 2739 . 2 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
4018, 27, 393sstr4d 3627 1 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  wss 3555  cmpt 4673   × cxp 5072  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850
This theorem is referenced by:  sibfof  30180
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